Chapter of his Clavis Mathematicae where he sayes that 43 7 is the Proportion of 31 to 7 for his meaning is that the Proportion of 43 7 to one is the Proportion of 31 to 7 whereas if he meant as you do then 86 7 should be double the Proportion of 31 to 7. Partly also because you think as in the end of the twentieth Proposition that if the Proportion of the Numerators of these Fractions to their Denominators decrease eternally they shall so vanish at last as to leave the Proportion of the sum of all the Squares to the sum of the greatest so often taken that is an infinite Number of times as one to three or the sum of the greatest to the sum of the increasing Squares as three to one for which there is no more reason then for four to one or five to one or any other such Proportion For if the Proportions come eternally nearer and nearer to the subtriple they must needs also come nearer and nearer to subquadruple and you may as well conclude thence that the upper Quantities shall be to the Lower Quantities as one to four or as one to five c. as conclude they are as one to three You can see without admonition what effect this false ground of yours will produce in the whole structure of your Arithmetica Infinitorum and how it makes all that you have said unto the end of your thirty-eighth Proposition undemonstrated and much of it false The thirty-nineth is this other Lemma In a Series of Quantities beginning with a Point or Cypher and proceeding according to the Series of the Cubique Numbers as 0. 1. 8. 27. 64 c. to finde the Proportion of the sum of the Cubes to the sum of the greatest Cube so many times taken as there be Terms And you conclude that they have the Proportion of 1 to 4 which is false Let the first Series be of three terms subscribed with the greatest the sum of the Cubes is nine the sum of all the greatest is 24 a quarter whereof is 6. But 9 is greater then 6 by three unities An unity is something Let it be therefore A. Therefore the Row of Cubes is greater then a quarter of three times eight by three A. Again let the Series have four terms as the sum of the Cubes is 36 a quarter of the sum of all the greatest is twenty-seven But thirty-six is greater then twenty-seven by nineâ— that is by 9 A. The excess therefore of the sum of the Cubes above the fourth part of the sum of all the greatest is increased by the increase of the Number of terms Again let the terms be five as the sum of the Cubes is one hundred the sum of all the greatest three hundred and twenty a quarter whereof is eighty But one hundred is greater then eighty by twenty that is by 20 A. So you see that this Lemma also is false And yet there is grounded upon it all that which you have of comparing Parabolas and Paraboloeides with the Parallelograms wherein they are accommodated And therefore though it be true that the Parabola is â…” and the Cubicall Paraboloeides ¾ of their Parallelograms respectively ' yet it is more then you were certain of when you referred me for the learning of Geometry to this Book of yours Besides any man may perceive that without these two Lemmas which are mingled with all your compounded Series with their excesses there is nothing demonstrated to the end of your Book Which to prosecute particularly were but a vain expence of time Truly were it not that I must defend my reputation I should not have shewed the world how little there is of sound Doctrine in any of your Books For when I think how dejected you will be for the future and how the grief of so much time irrecoverably lost together with the conscience of taking so great a stipend for mis-teaching the young men of the University the consideration of how much your friends wil be ashamed of you will accompany you for the rest of your life I have more compassion for you then you have deserved Your Treatise of the Angle of Contact I have before confuted in a very few leaves And for that of your Conique Sections it is so covered over with the scab of Symboles that I had not the patience to examine whether it be well or ill demonstrated Yet I observed thus much that you find a Tangent to a Point given in the Section by a Diameter given and in the next Chapter after you teach the finding of a Diameter which is not artificially done I observe also that you call the Parameter an Imaginary Line as if the place thereof were less determined then the Diameter it self and then you take a mean Proportionall between the intercepted Diameter and its contiguous ordinate Line to find it And t is true you find itâ— But the Parameter has a determined Quantity to be found without taking a mean Proportional For the Diameter and half the Section being given draw a Tangent through the Vertex and dividing the Angle in the midst which is made by the Diameter and Tangent the Line that so divideth the Angle will cut the crooked Line From 〈◊〉 intersection draw a Line if it be a Parabola Parallel to the Diameter and that Line shall cut off in the Tangent from the Vertex the Parameter sought But if the Section be an Eâ—lipsis or an Hyperbole you may use the same Method saving that the Line drawn from the intersection must not be Parallel but must pass through the end of the transverse Diameter and then also it shall cut off a part of the Tangent which measured from the Vertex is the Parameter So that there is no more reason to call the Parameter an Imaginary Line then the Diameter Lastly I observe that in all this your new Method of Coniques you shew not how to find the Burning Points which writers call the Foci and Umbilici of the Section which are of all other things belonging to the Coniques most usefull in Philosophy Why therefore were they not as worthy of your pains as the rest for the rest also have already been demonstrated by others You know the Focus of the Parabola is in the Axis distant from the Vertex a quarter of the Parameter Know also that the Focus of an Hyperbole is in the Axis distant from the Vertex as much as the Hypotenusall of a rectangled Triangle whose one side is half the transverse Axis the other side half the mean Proportionall between the whole transverse Axis and the Parameter is greater then half the transverse Axis The cause why you have performed nothing in any of your Books saving that in your Elenâ—…hus you have spied a few negligences of mine which I need not be ashamed of is this that you understood not what is Quantity Line Superâ—…ies Angle and Proportion without which you cannot have the Science
suppose that the crocked Line AB in the seventh Figure were not an Arch of a Circle do you think that the Angles which it maketh with the Subtense AB at the Points A and B must needs be equall Or if they be not does the excess of the Superficies of the Circle upon AD above the Superficies of the Cone or the exceis of the Superficiâ—s of the Portion of the Conoeides above the Superficies of the same Cone consist in the Angle DAB oâ— rather in the â—…tude of the two unequall Angles DAB and ABA You should have drawn some other crooked Line and made Tangents to it through A and B and you would presently have seen your error See how you can answer this for if this Demonstration of minestand firm I may be bold to say though the same be well Demonstrated by Arâ—… that this way of mine is more naturall as proceeding immediately from the naturall efficient causes of the effect contained in the conclusion and besides more brief and more easie to be followed by the fancy of the Reader To the fourteenth Article you say that I commit a Circle in that I require in the fourth Article the finding of two mean Proportionals and come not till now to show how it is to be done Nor now neither But in the mean time you commit two mistakes in saying so The place cited by you in the fourth Article is in the Latine Pag. 149. Lin 9. in the English Pag. 188. Lin. 3. Let any Reader judge whether that be a requiring it or a supposing it to be done this is your first mistake The second is that in this place the Preposition itself which is If those Deficient Figures could be described in a Parallelâ—gram exquisitely there might be found thereby between any two Lines given as many mean Proportionals as one would is a Theoreme upon supposition of these crooked Lines exquisitely drawn but you take it for a Probleme And proceeding in that error you undertake the invention of two mean Proportionals using therein my first Figure which is of the same construction with the eighth that belongeth to this fourteenth Article Your construction is Let there be taken in the Diameter CA Figure 1 the two given Lines or two others Proportionall to them as CH CG and their â—â—â—nate Lines HF GE which by construction are in subtriplicate Prâ—pâ—rtion of the intercepted Diameters These Lines will shew the Proportions which those four Proportionals are to 〈◊〉 But how will you find the Length of HF or GE the ordinate Lines Will you not do it by so drawing the crooked Line CFE as it may paâ—s through both the Points F and E You may make it pass through one of them but to make it pass through the other you must finde two mean Proportionals between GK and GL or between HI and HP Which you cannot do unless the crooked Line be exactly drawn which it cannot be by the Geometry of Plunes Go show this Demonstration of yours to Orontius and see what he will say to it I am now come to an end of your objections to the seventeenth Chapter where you have an Epiphonema not to be passed over in silence But becausâ— you pâ—etend to the Dâ—…tration of some of these Propositions by another Method in your Arithmetica Infinitorum I shall first try whether you be able to defend those Demonstrations as well as I have done theie of mine by the Method of Motion The first Proposition of your Arithmetica Infinitorum is this Lâ—mma In a Sâ—ries or Row of Quantities Arith â—tically Proportionall beginning at a Poiâ—t or Cypâ—â—r as 0 1 2 3 4 c. to finde the Proportion of the Aggregate of them all to the Aggregate of so many times the greatest as there are Terms This is to be done by multiplying the greatest into half the Number of the Terms The Demonstration is easie But how do you demonstrate the same The most simple way say you of finding this and some other Problemes it to do the thing it self a little way and to observe and compare the appearing Proportions and then by Induction to conclude it universally Egregious Logicians and Geometricians that think an Induction without a Numeration of all the particulars sufficient to infer a Conclusion universall and fit to be received for a Geometricall Demonstration But why do you limit it to the naturall consequution of the Numbers 0 1 2 3 4 c Is it not also true in these Numbers 0 2 4 6 c. or in these 0 7 14 21 c Or in any Numbers where the Diffâ—rence of nothing and the first Number is equall to the difference between the first and second and between the second and third c Again are not these Quantities 1 3 5 7 c. in continuall Proportion Arithmaticall And if you put before them a Cypher thus 0 1 3 5 7 do you think that the sum of them is equall to the half of five times seven Therefore though your Lemma be true and by me Chap. 13. Art 5. demonstrated yet you did not know why it is true which also appears most evidently in the first Proposition of your Conique-sections Where first you have this That a Parallelogram whose Altitude is infinitely little that is to say none is scarce any thing else but a Line Is this the Language of Geometry How do you determine this word scarce The least Altitude is Somewhat or Nothing If Somewhat then the first character of your Arithmeticall Progression must not be a cypher and consequently the first eighteen Propositions of this yâ—ur Arithmetica Infinitorum are all naught If Nothing then your whole figure is without Altitude and consequently your Understanding naught Again in the same Proposition you say thus We will sometimes call those Parallelograms rather by the name of Lines then of Parallelograms at least when there is no consideration of a determinate Altitude But where there is a consideration of a determinate Altitude which will happen sometimes there that little Altitude shall be so far considered as that being infinitely multiplyed it may be equall to the Altitude of the whole Figure See here in what a confusion you are when you resist the truth When you consider no determinate Altitude that is no Quantity of Altitude then you say your Parallelogram shall be called a Line But when the Altitude is determined that is when it is Quantity then you will call it a Parallelogram Is not this the very same doctrine which you so much wonder at and reprehend in me in your objections to my eighth Chapter and your word considered used as I used it 'T is very ugly in one that so bitterly reprehendeth a doctrine in another to be driven upon the same himself by the force of truth when he thinks not on 't Again seeing you admit in any case those infinitely little altitudes to be quantity what need you this limitation of yours so far forth as that by
are incongruous or a crooked and a straight line touch one another the contact is not in a Line but only in one Point and then your instance of a Circle and a Parabola was a wilfull cavill not befitting a Doctor If you either read them not or unstood them not it is your own fault In the rest that followeth upon this Article with your Diagram there is nothing against me nor any thing of use novelty subtilty or learning At the seventh Article where I define both an Angle simply so called and an Angle of Contingence by their severall generations namely that the former is generated when two straight Lines are coincident and one of them is moved and distracted from the other by circular motion upon one common Point resting c. You ask me to which of these kinds of Angle I refâ—r the Angle made by a straight Line when it cuts a crooked Line I answer easily and truly to that kind of Angle which is called simply an Angle This you understand not For how will you say can that Angle which is generated by the divergence of two straight Lines be other then Rectilineall O how can that Angle which is not comprehended by two straight Lines be other then Curvilineall I see what it is that troubles you namely the same which made you say before that if the Body which describes a Line be a Point then there is nothing which is not moved that can be called a Point So you say here If an Angle be generated by the motion of a straight Line then no Angle so generated can be Curvilineall Which is as well argued as if a man should say the House was built by the carriage and motion of Stone and Timber therefore when the carriage and that motion is ended it is no more a house Rectilineall and Curvilineall hath nothing to do with the nature of an Angle simply so called though it be essentiall to an Angle of Contact The measure of an Angle simply so called is a circumference of a Circle and the measure is alwayes the same kind of Quantity with the thing measured The Rectitude or Curvity of the Lines which drawn from the Center intercept the Arch is accidentary to the Angle which is the same whether it be drawn by the motion circular of a streight line or of a crooked The Diameter and the Circumference of a Circle make a right Angle and the same which is made by the Diameter and the Tangent And because the point of Contact is not as you think nothing but a line unreckoned and common both to the Tangent and the Circumference the same Angle computed in the Tangent is Rectilineall but computed in the Circumference not Rectilineall but mixt or if two Circles cut one another Curvilineall For every Chord maketh the same Angle with the Circumference which it maketh with the line that toucheth the Circumference at the end of the Chord And therefore when I divide an Angle simply so called into Rectilineall and Curvilineall I respect no more the generation of it then when I divide it into Right and Oblique I then respect the generation when I divide an Angle into an Angle simply so called and an Angle of Contact This that I have now said if the Reader remember when he reads your objections to this and to the nineth Article he will need no more to make him see that you are utterly ignorant of the nature of an Angle and that if ignorance be madness not I but you are mad and when an Angle is comprehended between a straight and a crooked Line if I may compute the same Angle as comprehended between the same straight Line and the Point of Contact that it is consonant to my definition of a Point by a Magnitude not considered But when you in your treatise de Angulo Contactûs Chap. 3. Pag. 6. Lin. 8. have these words Though the whole concurrent Lines incline to one another yet they form no Angle any where but in the very point of concourse You that deny a Point to be any thing tell me how two nothings can form an Angle or if the Angle be not formed neither before the concurrent Lines meet not in the Point of concourse how can you apprehend that any Angle can possibly be framed But I wonder not at this absurdity because this whole treatise of yours is but one absurdity continued from the beginning to the end as shall then appear when I come to answer your objections to that which I have briefly and fully said of that Subject in my 14. Chapter At the twelfth Article I confess your exception to my universall definition of Parallels to be just though insolently set down For it is no fault of ignorance though it also infect the demonstration next it but of too much security The Definition is this Parallels are those Lines or Superficies upon which two straight Lines falling and wheresoever they fall making equall Angles with them both are equall which is not as it stands universally true But inserting these words the same way and making it stand thus Parallel Lines or Superficies are those upon which two straight Lines falling the same way and wheresoever they fall making equall Angles are equall it is both true and universall and the following Consectary with very little change as you may see in the translation perspicuously demonstrated The same fault occurreth once or twice more and you triumph unreasonably as if you had given therein a very great proof of your Geometry The same was observed also upon this place by one of the prime Geometricians of Paris and noted in a Letter to his friend in these words Chap. 14. Art 12. the Definition of Parallels wanteth somewhat to be supplyed And of the Consectary he says it concludeth not because it is grounded on the Definition of Parallels Truely and severely enough though without any such words as savour of Arrogance or of Malice or of the Clown At the thirteenth Article you recite the Demonstration by which I prove the Perimeters of two Circles to be Proportionall to their Semidiameters and with Esto fortasse recte omnino noddying to the severall parts thereof you come at Length to my last inference Therefore by Chap. 13. Art 6. the Perimeters and Semidiameters of Circles are Proportionall which you deny and therefore deny because you say it followeth by the same Ratiocination that Circles also and Spheres are Proportionall to their Semidiameters For the same distance you say of the Perimeter from the Center which determines the magnitude of the Semidiameter determines also the magnitude both of the Circle and of the Sphere You acknowledge that Perimeters and Semidiameters have the cause of their determination such as in equal times make equall spaces Suppose now a Sphere generated by the Semidiameters whilst the Semicircle is turned about There is but one Radius of an infinite number of Radii which describes a great Circle all the